Heuristic approximations and abbreviations in the error backpropagation calculation process for SNNs are also major factors that compose the network learning paradigm. In general, learning methods using surrogate gradients ignore the error backpropagation paths led by resets of membrane potential, what is called reset path, in multi-spike [1]}, [2]}, [3]}. Backpropagating through surrogate gradient results in the fact that the reset information is propagated recursively by a number of times equal to the number of time steps, even though the actual reset is instantaneous at the spike timing. Even though the amount of the approximation error induced by a single surrogate derivative is tolerable, the accumulated error can be huge because the number of time steps is usually by far larger than the number of spikes [1]}, [3]}. To avoid this approximation error diverging is the main reason why they ignore the reset paths. On the other hand, the timing-based method used in this study does not use any approximation in the whole calculation process of backpropagation, i.e., it realizes an exact gradient calculation. It should be noted, however, that the timing-based method cannot capture the occurrence and disappearance of sudden spikes caused by weight updates and the accompanying changes in the number of spikes. The sudden spike changes pointed out here are not spike changes that increase or decrease with continuous changes in membrane potential in the time direction around the end time \(t_{out}\) , but spike changes that occur with discontinuous membrane potential changes in the middle of a given period \((0, t_{out})\) . Although this is a challenge inherent to the timing-based method as pointed out by SpikeProp [6]}, it was experimentally shown that learning works well both for single-spike SNNs and also for the multi-spike SNNs in this study. The impact of this sudden spike variation is one of the issues that should be considered in the future.
